# Minimum value on an open continuous function

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## Homework Statement

Suppose that f is a continuous function on (a,b) and $lim_{x \rightarrow a^{+}} f(x) = lim_{x \rightarrow b^{-}} f(x) = \infty.$ prove that f has a minimum on all of (a,b)

## The Attempt at a Solution

I have not tried an actual attempt yet. The only think I can think of doing is making two sequences that approach a common point on the domain of f. One sequence starting at a, and the other starting at b. Then show that the range of these sequences is decreasing and tends to the same value. This seems a bit too complicated to me for such a problem.

I am interested in where to start. Logically, it makes sense to me that there should be a minimum. I just don't know how to explain it using math.

Thanks.